Symplectic Model-Reduction with a Weighted Inner Product
This work extends structure-preserving model reduction to problems with non-Euclidean inner products, which is important for applications in mechanics and physics.
The authors generalize symplectic model reduction to accommodate weighted inner products, preserving Hamiltonian structure while improving accuracy and stability. They demonstrate the method on a vibrating elastic beam and the sine-Gordon equation.
In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems, symplectic model reduction seeks to construct a reduced system that preserves the symplectic symmetry of Hamiltonian systems. However, symplectic methods are based on the standard Euclidean inner products and are not suitable for problems equipped with a more general inner product. In this paper, we generalize symplectic model reduction to allow for the norms and inner products that are most appropriate to the problem while preserving the symplectic symmetry of the Hamiltonian systems. To construct a reduced basis and accelerate the evaluation of nonlinear terms, a greedy generation of a symplectic basis is proposed. Furthermore, it is shown that the greedy approach yields a norm-bounded reduced basis. The accuracy and the stability of this model reduction technique are illustrated through the development of reduced models for a vibrating elastic beam and the sine-Gordon equation.